Fame For Styria 2014: Pure Cubic Fields 1

Pure Cubic Fields 1

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Pure Cubic Fields 1

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4. Ambiguous principal ideals.

Since the cyclotomic unit group Uk is generated by < -1, ζ, >,
and since NormN|kUN ≥ NormN|kUk = Uk3 = < -1 >,
we obtain bounds for the unit norm index
(Uk:NormN|kUN) ∈ {1,3},
according to whether ζ ∈ NormN|kUN or not.

The somewhat abstract quotient EN|k/UN1 - σ
is isomorphic to the more ostensive quotient PNGal(N|k)/Pk
of the group of ambiguous principal ideals of N|k
modulo the subgroup of principal ideals of k,
according to Iwasawa (or also to Hilbert's Theorems 92 and 94),
and by the Takagi/Hasse Theorem on the Herbrand quotient, we have
(PNGal(N|k):Pk) = (EN|k:UN1 - σ) = 3 (Uk:NormN|kUN) ∈ {3,9}.

Even in the worst case (PNGal(N|k):Pk) = 3,
there are at least the radicals D1/3, D2/3, and the unit 1
which generate 3 distinct ambiguous principal ideals of L|Q.

Examples 4.1.
We give the smallest radicands D of pure cubic fields L = Q(D1/3)
where the two values of #Pr = (PNGal(N|k):Pk) actually occur:
#Pr = 3 for D = 3 of type γ,
#Pr = 9 for D = 2 of type β.

5. Differential principal factorizations (DPF).

As opposed to Hilbert's Theorem 94, the subgroup
PN ∩ Ik/Pk ≤ PNGal(N|k)/Pk,
the so-called capitulation kernel of N|k is trivial, because h(k) = 1.
However, the elementary abelian 3-group Pr = PNGal(N|k)/Pk,
whose generators are principal ideals dividing the different of N|k
(so-called differential principal factors),
consists of two nested subgroups, Pr ≥ Pa
(similar to but not identical with our definitions in [Ma2]),
  • absolute DPF of L|Q, Pa, always containing the above mentioned radicals,
  • relative DPF of N|k, Pr \ Pa, such that the proper cosets of Pa in Pr
    do not project down to L|Q by taking the norm.
The existence, resp. the lack, of certain prime divisors of the conductor f
permits some criteria for the classification of pure cubic fields.

Theorem 5.1.
  1. (Uk:NormN|kUN) = 1 can occur only when f is divisible by no other primes than
    3 and / or primes qj ≡ ±1 (mod 9).
  2. Relative DPF of N|k can occur only if some prime qj ≡ +1 (mod 3) divides f.
  3. Absolute DPF of L|Q, distinct from radicals, must exist when
    no prime ≡ +1 (mod 3) divides f
    and f has at least one prime factor distinct from 3 and from primes ≡ -1 (mod 9).

Remarks 5.1.
1. The condition for ζ ∈ NormN|kUN is due to
the properties of the cubic Hilbert symbol (third power norm residue symbol) over k.
2. The condition for relative DPF of N|k is a consequence of the decomposition law
(e,f,g) = (1,1,2) for primes ≡ +1 (mod 3) in k.

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Trade, Science, Art and Industry
Daniel C. Mayer
Principal investigator of the
International Research Project
Towers of p-Class Fields
over Algebraic Number Fields
supported by the Austrian Science Fund (FWF):
P 26008-N25

Bibliographical References:

[Ma2] D. C. Mayer,
Classification of dihedral fields,
University of Manitoba, Winnipeg, Manitoba, Canada, 1991.

Web master's e-mail address:

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